.999... = 1

[Snorri1234]

Damn. I mean, I know that Prowler was just trolling, I just never figured that other morons would agree with him.

[e_i_pi]

But, as Sully has already pointed out, 0.999... is not a series. It is a single number.

Not a problem. Can you be more rigorous in your proof and write out for me 0.999... in its entirety

[a.sub]

But, as Sully has already pointed out, 0.999... is not a series. It is a single number.

Not a problem. Can you be more rigorous in your proof and write out for me 0.999... in its entirety

can you write out .333... in its full form?
no
you can only write it as a fraction 1/3
because ALL repeating decimals can be
now whats teh fraction of .999...?????

[john9blue]

I think someone brought this up before.

Given: x = .999... (this is not false, it's just a starting point)
10x = 9.999... by shifting the decimal point
10x - x = 9x by easy algebra subtraction
9.999... - .999... = 9 by subtracting each decimal place except the tens
9x = 9 by seeing that the left sides of the above two equations are equal, because of the first two equations
x = 1 by dividing both sides by 9

Therefore, .999... = 1 (because the first and last equations have identical left sides).

There's nothing wrong with this proof.

Just for fun, let's make a list.

People who think that .999... = 1
john9blue
Timminz
owheelj
a.sub
sully800
SultanOfSurreal
Snorri1234
xelabale

People who think otherwise
e_i_pi
TheProwler
KLOBBER
Suspect101

Then we will know whom to ridicule once this debate has been decided.

[MeDeFe]

You can add me to the "People who think that..." list.

[chipv]

0.1111111... = 1/9
0.2222222... = 2/9
...
0.9999999... = 9/9 = 1

This takes some swallowing so here is a better 'proof' using infinite series:

0.9999... = 0.9 x (1 + 1/10 + 1/100 + ...)

The bracketed portion is an infinite geometric series.
The sum of a geometric series is finite as long as the terms approach zero (which it does in this case).

The first term is 1 and the ratio is 1/10

The sum of the series is

First Term / (1 - ratio)

= 1 / (1 - 1/10) = 1/ (9/10) = 10/9

So 0.9999 = 0.9 x 10/9 = 1

[Timminz]

Nice!

What's that now? 3? 4 perfectly valid proofs?

[john9blue]

truth

The post heard round the world.

[AceArtemis]

Ok, I didn't read the entire thread, so I don't know if this argument has been presented already, but here goes:

1. If two real numbers are not equal to each other, there are an infinite number of real numbers between them on the number line.
2. There are no real numbers between 0.999... and 1 on the number line. (To dispute this step, simply name ONE real number between 0.999... and 1 on the number line)
3. Therefore, the statement that 0.999... and 1 are not equal to each other is false.
4. If two numbers are not not equal to each other, they are equal to each other.
5. Therefore, 0.999...=1.

In my opinion, this should be a question on the SAT:
0.999...=1
a. true
b. false

[Timminz]

And another excellent proof.

[e_i_pi]

Nice!

What's that now? 3? 4 perfectly valid proofs?

I think the number you're looking for is infinity plus one.

I don't buy these proofs, I haven't bought them for 20 years, I'm not buying them now. What gets me is the hypocricy of argument that goes on when you start questioning this. I proved that 0.999... != 1 using infinite series, am told that I can't use infinite series, and then chipv uses an infinite series to prove 0.999... = 1, and that counts as above board.

I have issues with "..." as notation - does not compute. What does "..." equal? It doesn't equal anything, it is by definition infinite, so how can you define equality on it? Infinity is not a number, it can only exist as a limit, so getting around that by saying "..." = infinity, then performing operations on "..." as if it is numerical is not valid.

As an example, I'll quote Timminz:

9.999... - 0.999... = 9.

Show me where that is incorrect.

Well, you're assuming that "..." is well-defined and can be using in arithmetic. Show me where "..." is defined numerically please. While you're at it, show me where x + infinity - infinity = x. Present that to your teachers at whatever university you go to, and watch them get out the red pen. Does not compute. Infinity, and all it's substitutions and derivatives are non-numerical, and not part of any group, therefore you cannot perform group operations upon them. Infinity exists as a limit, and is used for theoretical purposes.

If you still think that the smoke and mirrors use of "..." to addle our common conceptions of how equality, equations, and mathematics works, I suggest you start reading Kurt Gödel.

[sully800]

1/11 = 0.090909...
10/11 = 0.909090.....

1/11 + 10/11 = 1

0.090909... + 0.909090... = 0.999999

1 = 0.999999

QED

Need more?

[e_i_pi]

1/11 = 0.090909...
10/11 = 0.909090.....

1/11 + 10/11 = 1

0.090909... + 0.909090... = 0.999999

1 = 0.999999

QED

Need more?

You're still ascribing numerical value to "..."

[sully800]

1/11 = 0.090909...
10/11 = 0.909090.....

1/11 + 10/11 = 1

0.090909... + 0.909090... = 0.999999

1 = 0.999999

QED

Need more?

You're still ascribing numerical value to "..."

Yes, and it's the same as saying recurring or a dash over the numbers or any other way of expressing that the decimal repeats infinitely, which is well understood by all of us. It has a real numerical value that carries on infinitely.

0.3333... or 0.3333recurring is not a mere representation of 1/3. It IS 1/3. Exactly equivalent. If they were not equivalent you would be able to tell me the difference between the two numbers, but there is literally no difference.

[Timminz]

1/11 = 0.090909...
10/11 = 0.909090.....

1/11 + 10/11 = 1

0.090909... + 0.909090... = 0.999999

1 = 0.999999

QED

Need more?

You're still ascribing numerical value to "..."

Only in relation to other "..."s.

[e_i_pi]

1/11 = 0.090909...
10/11 = 0.909090.....

1/11 + 10/11 = 1

0.090909... + 0.909090... = 0.999999

1 = 0.999999

QED

Need more?

You're still ascribing numerical value to "..."

Yes, and it's the same as saying recurring or a dash over the numbers or any other way of expressing that the decimal repeats infinitely, which is well understood by all of us. It has a real numerical value that carries on infinitely.

0.3333... or 0.3333recurring is not a mere representation of 1/3. It IS 1/3. Exactly equivalent. If they were not equivalent you would be able to tell me the difference between the two numbers, but there is literally no difference.

You can't simply say 0.333... = 1/3 without giving proof. It is a representation, it is not a proof. Saying that it is true does not prove it is true, otherwise I could say the world is flat. You need to define "..." Now, the only mathematical way to define "..." is to say that it is an infinite series of smaller and smaller decimals. If it is an infinite series, then you can prove 0.999... = 1 and 0.999... != 1, as has been shown above. The correct answer to this question is "mu".

[TheProwler]

I understand limits thanks.

You still haven't answered my question - the last digit is a 2 or a 4? or is one of the 3s worth less than the corresponding 3?

Those the options - 2 or a 4, or a 3 worth less than another 3. Which is it?

I'm not going to shake hands, I'm going to keep arguing until you give up or until you admit that I'm right. This has nothing to do with me believing other people, it's to do with me being able to calculate 3x3 as well as 1/3 and 9/3. Merely with those abilities I can work out for myself that 1=0.999...

It's funny that you state that 1/3 does not equal 0.333... What does it equal then? Work it out on a bit of paper. 1/3 - first digit 0. Decimal point. Next digit 3. Next digit 3. Next digit 3 etc. At what point do you realise that it's going to be 0 followed by a decimal point followed by an infinite number of threes. Which is obviously expressed as 0.3 recurring. If we go along that line of infinite threes and multiple each individual digit by 3 what do we get? A 9 every time - all the way down the line. At no point do we have a reason to stop writing down nines. Therefore we *know* that 0.333... x 3 = 0.999... We also know that 0.333... = 1/3. and we know what we get when we multiply 1/3 with 3. If any of these equations are incorrect, don't tell me what my mistake was, tell me what the actual correct answer is.

I hope I don't get sucked into answering every post made in the last 4 pages or so...

You said "Which is obviously expressed as 0.3 recurring."

The deficiency is in the method that we have in expressing this number.

Think of your turtle and man walking idea...say they each continue, each at a constant speed but one slower than the other, for an infinite amount of time. How far do they each go? An infinite distance. But who has gone further? Well, the man goes, say, 10 times as fast. At any point in time, he has gone ten times as far. So does infinity equal infinity? Hmm...well, an infinite is only approached, in reality. In one case, it is approached 10 times faster.... Do they go the same distance? Never. But you are under the impression that when infinity is "reached", it is "reached" at the same time....hmmmm? Has the man, who has traveled 10 times further not gone further? If there are an infinite number of 3's following a decimal place, but in one expression, the 3's are tallied just a little faster, is that number just a little bigger than the one where the 3's are tallied slower? Hmm....maybe there is a deficiency is just expressing the number in terms of 0.333recurring.

Think of an infinitely small point at coordinates 10,10. You might draw that with a dot. But the dot is infinitely too large. If we make your dot 1/2 the size and repeat this process infinitely, in an effort to "zero-in" on the point, does the dot ever get to where it needs to be? Does it ever get infinitely small? Never. Does the point actually exist? Yes it does...right there at 10,10. 1-0.999recurring is pretty close to the size of the point. But it can't be zero. If it were zero, the point wouldn't exist. But it does. Right there. At 10,10.

When we depict 1/3 with this expression: 0.333recurring, and we depict 0.999recurring/3 with this expression: 0.333recurring, we are demonstrating the deficiency in the way we express a two numbers that are different - because we express them the same way.

Here's something to think about: 0.999recurring does not exist in reality. 1 does exist in reality. Does 0.333recurring exist? Well, if it is used to represent 1/3, then sure, it does exist (because we know it is being used to express a value that exists in reality). If it is used to express 0.999recurring/3, it does not exist in reality. It is the nature of the universe. At least our understanding of it. We have all forged ahead, just accepting the idea that 0.999recurring exists. But it does not. Not in our understanding of the universe.

Can you write an equation without using recursion or the use of an infinite series to produce an answer of 0.999recurring? If you can, check your work.

[a.sub]

e_i_pi
i challenge you to argue this one

In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted (for quotient).

The decimal expansion of a rational number always either terminates after finitely many digits or begins to repeat the same sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.

.9999999999...
is a repeating decimal as we have defined it
ergo
according to the DEFINITION underlined in the section above in blue
then it is a rational number
ergo
according to the DEFINITION underlined in the section above in red

The above section if fact as defined by the very basic laws of abstract algebra and the concept of numerical definitions that all numbers must follow

so lets find this god damned fraction, if it doesnt equal 1 i will very happily step down and admit im wrong

[TheProwler]

If .333... = 1 and .333... = .999..., then .999... = 1. QED,MF.

I'll ignore the mistake...I know what you meant.

What mistake? You're saying that .333... = 1/3, but it's a different .333... than what you're talking about.

Okay, so you meant ".333...=1 and .333...=.999..."

My bad...

[TheProwler]

i found his mistake
[

You are failing to understand that limits can be approached at different rates. Did I mention that there is a deficiency in the way we express recursive numbers?

[TheProwler]

Infinity is only a theory after all.

Give this man a Gold Star. Come join me (and a few others) at the front of the class.

[TheProwler]

Infinity is only a theory after all.

So is 1.

I thought I ate 1 grape earlier this evening. Goddamn free particles! One got away!



Errr...I mean "1" got away...

[TheProwler]

Agreed, this isn't a matter of opinion - you'll find it in plenty of text books etc.

I would like you to list ANY text book, edition, and page number that claims .99999 = 1.

It's normally referred to on Page 9.9999recurring.



(just turn to Page 10)

[TheProwler]

Agreed, this isn't a matter of opinion - you'll find it in plenty of text books etc.

I would like you to list ANY text book, edition, and page number that claims .99999 = 1.

nope, i found one. I am an idiot

I still disagree because I found the same problem with this proof. Two numbers are considered equal if their difference is 0.
If you try and take 1 and subtract .999.... you never get an answer. It repeats forever and will never stop, therefore you will never get a difference but you never get 0 either.

You are trying to solve for a solution for an unbounded equation, which gives you a theory as to what it is, not what it actually is. I have found what you are talking about, but they still "assign to the notation "0.999…" is defined to be the real number which is the limit of the convergent sequence (0.9, 0.99, 0.999, 0.9999, …)." The limit of a convergent sequence never reachs its limit. it only gets infinteciamly closer to it.

I still believe the key here is convergence, asymptotes, and limits.

And this brings up an interesting thought....

Once an idiot, not always an idiot.

To the head of the class for you too!

[TheProwler]

A law of mathmatics is: If you take a finite number and subtract an infinte number you get another infinte number.

Like if you take 1 (finite)-.333333...(infinite).... = .6666.....(infinte)

If you take 1 - .999999 you do not get 0, because 0 is finite, you get 0.000000.... which is infinite.

Oh oh, you found a law of mathematics that contradicts a text book.

Is there a mathematical lawyer in the house?